I was asked to draw find the angles of a triangle with angles with the ratio of 10:7 and 7:3.
Multiplying all these numbers by nine gives angles of $90,63,$ and $27$. How can I now find the ratio of the sides?
This is my work: $$\frac{a}{\sin{\left(90\right)}}=\frac{b}{\sin{\left(63\right)}}=\frac{c}{\sin{\left(27\right)}}\\ a=\frac{b}{.891}=\frac{c}{.45399}$$ Ratio of the sides: $$1:0.891 \text{ and } 0.891:0.45399$$
Does this make sense?
You logic is sound because one Pythagorean triple that fits your picture is $$(589,300,661)$$
where $\sin^{-1}\big(\frac{300}{661}\big)\approx 0.471089961 r\approx 26.99146656^\circ$
Side ratios: $$A/C= 589/661\approx 0.89107413$$ $$B/C= 300/661\approx 0.453857791$$ $$A/B=589/300\approx 1.963333333$$